Interpreting and using heterogeneous choice

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Interpreting and using heterogeneous choice
generalized ordered logit models

• Richard Williams
• Department of Sociology
• University of Notre Dame
• July 2006
• http//www.nd.edu/rwilliam/

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The gologit/gologit2 model

• The gologit (generalized ordered logit) model can
  be written as
• The unconstrained model gives results that are
  similar to running a series of logistic
  regressions, where first it is category 1 versus
  all others, then categories 1 2 versus all
  others, then 1, 2 3 versus all others, etc.
• The unconstrained model estimates as many
  parameters as mlogit does, and tends to yield
  very similar fits.

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• The much better known ordered logit (ologit)
  model is a special case of the gologit model,
  where the betas are the same for each j (NOTE
  ologit actually reports cut points, which equal
  the negatives of the alphas used here)

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• The partial proportional odds models is another
  special case some but not all betas are the
  same across values of j. For example, in the
  following the betas for X1 and X2 are constrained
  but the betas for X3 are not.

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Key advantages of gologit2

• Can estimate models that are less restrictive
  than ologit (whose assumptions are often
  violated)
• Can estimate models (i.e. partial proportional
  odds) that are more parsimonious than non-ordinal
  alternatives, such as mlogit
• HOWEVER, there are also several potential
  concerns users may not be aware of or have not
  thought about

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Concern 1 Unconstrained model does not require
ordinality

• As Clogg Shihadeh (1994) point out, the totally
  unconstrained model arguably isnt even ordinal
• You can rearrange the categories, and fit can be
  hardly affected
• If a totally unconstrained model is the only one
  that fits, it may make more sense to use mlogit
• Gologit is mostly useful when you get a
  non-trivial of constraints.

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Concern II Estimated probabilitiescan go
negative

• Unlike other categorical models, estimated
  probabilities can be negative.
• This was addressed by McCullaph Nelder,
  Generalized Linear Models, 2nd edition, 1989, p.
  155The usefulness of non-parallel regression
  models is limited to some extent by the fact that
  the lines must eventually intersect.  Negative
  fitted values are then unavoidable for some
  values of x, though perhaps not in the observed
  range.  If such intersections occur in a
  sufficiently remote region of the x-space, this
  flaw in the model need not be serious.

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• Probabilities might go negative in unlikely or
  impossible X ranges, e.g. when years of education
  is negative or hourly wages are gt 5 million.
• But, it could also happen with more plausible
  sets of values
• Multiple tests with 10s of thousands of cases
  typically resulted in only 0 to 3 negative
  predicted probabilities.
• Seems most problematic with small samples,
  complicated models, analyses where the data are
  being spread very thin
• they might be troublesome regardless - gologit2
  could help expose problems that might otherwise
  be overlooked
• Can also get negative predicted probabilities
  when measurement of the outcome isnt actually
  ordinal

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• gologit2 now checks to see if any in-sample
  predicted probabilities are negative.
• It is still possible that plausible values not
  in-sample could produce negative predicted
  probabilities.
• You may want to use some other method if there
  are a non-trivial number of negative predicted
  probabilities and you are otherwise confident in
  your models and data.

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Concern III How do youinterpret the results???

• One rationale for ordinal regression models is
  that there is an underlying, continuous y that
  reflects the dependent variable we are interested
  in.
• y is unobserved, however. Instead, we observe
  y, which is basically a collapsed/grouped version
  of the unobserved y.
• High Income, Moderate Income and Low Income are a
  collapsed version of a continuous Income variable
• Some ranges of attitudes can be collapsed into a
  5 category scale ranging from Strongly Disagree
  to Strongly Agree
• As individuals cross thresholds (aka cut-points)
  on y, their value on the observed y changes

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• Question What does the gologit model mean for
  the behavior we are modeling? Does it mean the
  slopes of the latent regression are functions of
  the left hand side variable, that there is some
  sort of interaction effect between x and y? i.e.
• y    beta1'x e  if y 1
• y    beta2'x e  if y 2

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• Further, does the whole idea of an underlying y
  go out the window once you allow a single
  non-proportional effect? If so, how do you
  interpret the model?
• In an ordered logit (ologit) model, you only have
  one predicted value for y
• But in a gologit model, once you have a single
  non-parallel effect, you have M-1 linear
  predictions (similar to mlogit)

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Interpretation 1 gologit as non-linear
probability model

• As Long Freese (2006, p. 187) point out The
  ordinal regression model can also be developed as
  a nonlinear probability model without appealing
  to the idea of a latent variable.
• Ergo, the simplest thing may just be to interpret
  gologit as a non-linear probability model that
  lets you estimate the determinants probability
  of each outcome occurring. Forget about the idea
  of a y
• Other interpretations, however, can preserve or
  modify the idea of an underlying y

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Interpretation 2 State-dependent reporting bias
- gologit as measurement model

• As noted, the idea behind y is that there is an
  unobserved continuous variable that gets
  collapsed into the limited number of categories
  for the observed variable y.
• HOWEVER, respondents have to decide how that
  collapsing should be done, e.g. they have to
  decide whether their feelings cross the threshold
  between agree and strongly agree, whether
  their health is good or very good, etc.

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• Respondents do NOT necessarily use the same frame
  of reference when answering, e.g. the elderly may
  use a different frame of reference than the young
  do when assessing their health
• Other factors can also cause respondents to
  employ different thresholds when describing
  things
• Some groups may be more modest in describing
  their wealth, IQ or other characteristics

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• In these cases the underlying latent variable may
  be the same for all groups but the
  thresholds/cut points used may vary.
• Example an estimated gender effect could reflect
  differences in measurement across genders rather
  than a real gender effect on the outcome of
  interest.
• Lindeboom Doorslaer (2004) note that this has
  been referred to as state-dependent reporting
  bias, scale of reference bias, response category
  cut-point shift, reporting heterogeneity
  differential item functioning.

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• If the difference in thresholds is constant
  (index shift), proportional odds will still hold
• EX Womens cutpoints are all a half point higher
  than the corresponding male cutpoints
• ologit could be used in such cases
• If the difference is not constant (cut point
  shift), proportional odds will be violated
• EX Men and women might have the same thresholds
  at lower levels of pain but have different
  thresholds for higher levels
• A gologit/ partial proportional odds model can
  capture this

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• If you are confident that some apparent effects
  reflect differences in measurement rather than
  real differences in effects, then
• Cutpoints (and their determinants) are
  substantively interesting, rather than just
  nuisance parameters
• The idea of an underlying y is preserved
  (Determinants of y are the same for all, but
  cutpoints differ across individuals and groups)
• You should change the way predicted values are
  computed, i.e. you should just drop the
  measurement parameters when computing predictions
  (I think!)

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• Key advantage This could greatly improve
  cross-group comparisons, getting rid of
  artifactual differences caused by differences in
  measurement.
• Key Concern Can you really be sure the
  coefficients reflect measurement and not real
  effects, or some combination of real
  measurement effects?

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• Theory may help if your model strongly claims
  the effect of gender should be zero, then any
  observed effect of gender can be attributed to
  measurement differences.
• But regardless of what your theory says, you may
  at least want to acknowledge the possibility that
  apparent effects could be real or just
  measurement artifacts.

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Interpretation 3 The outcome ismulti-dimensional

• A variable that is ordinal in some respects may
  not be ordinal or else be differently-ordinal in
  others. E.g. variables could be ordered either
  by direction (Strongly disagree to Strongly
  Agree) or intensity (Indifferent to Feel Strongly)

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• Suppose women tend to take less extreme political
  positions than men.
• Using the first (directional) coding, an ordinal
  model might not work very well, whereas it could
  work well with the 2nd (intensity) coding.
• But, suppose that for every other independent
  variable the directional coding works fine in an
  ordinal model.

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• Our choices in the past have either been to (a)
  run ordered logit, with the model really not
  appropriate for the gender variable, or (b) run
  multinomial logit, ignoring the parsimony of the
  ordinal model just because one variable doesnt
  work with it.
• With gologit models, we have option (c)
  constrain the vars where it works to meet the
  parallel lines assumption, while freeing up other
  vars (e.g. gender) from that constraint.

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• This interpretation suggests that there may
  actually be multiple ys that give rise to a
  single observed y
• NOTE This is very similar to the rationale for
  the multidimensional stereotype logit model
  estimated by slogit.

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Interpretation 4 The effect of x on y does
depend on the value of y

• There are actually many situations where the
  effect of x on y is going to vary across the
  range of y.
• EX A 1-unit increase in x produces a 5 increase
  in y
• So, if y 10,000, the increase will be 500.
  But if y 100,000, the increase will be 5,000.

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• If we were using OLS, we might address this issue
  by transforming y, e.g. takes its log, so that
  the effect of x was linear and the same across
  all values of the transformed y.
• But with ordinal methods, we cant easily
  transform an unobserved latent variable so with
  gologit we allow the effect of x to vary across
  values of y.
• This suggests that there is an underlying y but
  because we cant observe or transform it we have
  to allow the regression coefficients to vary
  across values of y instead.

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• Substantive example Boes Winkelman,
  2004Completely missing so far is any evidence
  whether the magnitude of the income effect
  depends on a persons happiness is it possible
  that the effect of income on happiness is
  different in different parts of the outcome
  distribution? Could it be that money cannot buy
  happiness, but buy-off unhappiness as a proverb
  says? And if so, how can such distributional
  effects be quantified?

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One last methodological noteon using gologit2

• Despite its name, gologit2 actually supports 5
  link functions logit, probit, log-log,
  complementary log-log, Cauchit. Each of these
  has a somewhat different distribution, differing,
  for example, in how heavy the tails are and how
  likely it is you will get extreme values.
• Changing the link function may change whether or
  not a variable meets the parallel lines
  assumption.
• Ergo, before turning to more complicated models
  and interpretations, you may want to try out
  different link functions to see if one of them
  makes it more likely that the parallel lines
  assumption will hold.

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An Alternative to gologit Heterogeneous Choice
(aka Location-Scale) Models

• Heterogeneous choice (aka location-scale) models
  can be generalized for use with either ordinal or
  binary dependent variables. They can be estimated
  in Stata by using Williams oglm program. (Also
  see handout p. 3). For a binary outcome,

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• The logit ordered logit models assume sigma is
  the same for all individuals
• Allison (1999) argues that sigma often differs
  across groups (e.g. women have more heterogeneous
  career patterns).
• Unlike OLS, failure to account for this results
  in biased parameter estimates.
• Williams (2006) shows that Allisons proposed
  solution for dealing with across-group
  differences is actually a special case of the
  heterogeneous choice model, and can be estimated
  (and improved upon) by using oglm.

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• Heterogeneous choice models may also provide an
  attractive alternative to gologit models
• Model fits, predicted values and ultimate
  substantive conclusions are sometimes similar
• Heterogeneous choice models are more widely known
  and may be easier to justify and explain, both
  methodologically theoretically

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Example

• (Adapted from Long Freese, 2006 Data from the
  1977 1989 General Social Survey)
• Respondents are asked to evaluate the following
  statement A working mother can establish just
  as warm and secure a relationship with her child
  as a mother who does not work.
• 1 Strongly Disagree (SD)
• 2 Disagree (D)
• 3 Agree (A)
• 4 Strongly Agree (SA).

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• Explanatory variables are
• yr89 (survey year 0 1977, 1 1989)
• male (0 female, 1 male)
• white (0 nonwhite, 1 white)
• age (measured in years)
• ed (years of education)
• prst (occupational prestige scale).

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• See handout pages 2-3 for Stata output
• For ologit, chi-square is 301.72 with 6 d.f. Both
  gologit2 (338.30 with 10 d.f.) and oglm (331.03
  with 8 d.f.) fit much better. The BIC test picks
  oglm as the best-fitting model.
• The corresponding predicted probabilities from
  oglm and gologit all correlate at .99 or higher.

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• The marginal effects (handout p. 4) show that the
  heterogeneous choice and gologit models agree
  (unlike ologit) that the main reason attitudes
  became more favorable across time was because
  people shifted from extremely negative positions
  to more moderate positions
• NOTE In Stata, marginal effects for multiple
  outcome models are easily estimated and formatted
  for output by using Williamss mfx2 program in
  conjunction with programs like estout and
  outreg2.
• oglm gologit also agree that it isnt so much
  that men were extremely negative in their
  attitudes it is more a matter of them being less
  likely than women to be extremely supportive.

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• In the oglm printout, the negative coefficients
  in the variance equation for yr89 and male show
  that there was less variability in attitudes in
  1989 than in 1977, and that men were less
  variable in their attitudes than women.
• This is substantively interesting and relatively
  easy to explain

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• Empirically, youd be hard pressed to choose
  between oglm and gologit in this case
• Theoretical issues or simply ease and clarity of
  presentation might lead you to prefer oglm
• However, see Williams (2006) and Keele Park
  (2006) for potential problems and pitfalls with
  the heterogeneous choice model
• Of course, in other cases gologit models may be
  clearly preferable

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For more information, see

• http//www.nd.edu/rwilliam/gologit2
• http//www.nd.edu/rwilliam/oglm/